# A Review of the Rules for Exponents

Rules for Exponents are used to simplify expressions with complex exponents. Below is List of Rules for Exponents and examples how to use these rules.

### Power Rule (Powers to Powers):

(a^m)^n = a^{mn}, this says that to raise a power to a power you need to multiply the exponents. There are several other rules that go along with the power rule, such as the product-to-powers rule and the quotient-to-powers rule.

### Negative Exponent Rule:

a^{-m}=\dfrac{1}{a^m} , this says that negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents.

### Product Rule:

a^m \cdotp a^n = a^{m + n}, this says that to multiply two exponents with the same base, you keep the base and add the powers.

### Quotient Rule

\dfrac{a^m}{a^n}=a^{m-n} , this says that to divide two exponents with the same base, you keep the base and subtract the powers. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents.

### Zero-Exponent Rule:

a^0 = 1, this says that anything raised to the zero power is 1. In fact, zero exponent rule can be taken as a special case of quotient rule when powers of numerator and denominator are equal.

Now that we have reviewed the rules for exponents, here are the steps required for simplifying exponential expressions (notice that we apply the rules in the same order the rule were written above):

**Step 1**: Apply the Zero-Exponent Rule. Change anything raised to the zero power into a 1.

**Step 2**: Apply the Power Rule. Multiply (or distribute) the exponent outside the parenthesis with every exponent inside the parenthesis, remember that if there is no exponent shown, then the exponent is 1.

**Step 3**: Apply the Negative Exponent Rule. Negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. Note that the order in which things are moved does not matter.

**Step 4**: Apply the Product Rule. To multiply two exponents with the same base, you keep the base and add the powers.

**Step 5**: Apply the Quotient Rule. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents and a repeat of step 3.

**Step 6**: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions.

### Example 1

Simplify the expression with exponent

(\dfrac{a^{-1}\sqrt{a} }{a^2} )^2

=\dfrac{a^{-2}\cdotp a }{a^{4}}

=\dfrac{a^{-1}}{a^4}

=\dfrac{1}{a^5}

Step 1: Apply the Zero-Exponent Rule. In this case, there are no zero powers.

Step 2: Apply the Power Rule.

Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa.

Step 4: Apply the Product Rule.

Step 5: Apply the Quotient Rule. In this case, the variables ended up in the denominator

Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does not reduce.

### Example 2

Simplify the expression with exponent

9^a\cdotp (\dfrac{1}{3} )^{2a-6}\cdotp 81^0

=9^a\cdotp (\dfrac{1}{3} )^{2a-6}

=9^a\cdotp \dfrac{1}{3^{2a-6}}

=9^a\cdotp \dfrac{1}{3^{2a}\cdotp 3^{-6} }

=9^a\cdotp \dfrac{3^{6}}{3^{2a} }

=9^a\cdotp \dfrac{3^{6}}{9^a}

=3^6

Step 1: Apply the Zero-Exponent Rule.

Step 2: Apply the Power Rule.

Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa.

Step 4: Apply the Product Rule. In this case, the product rule does not apply.

Step 5: Apply the Quotient Rule. In this case, the quotient rule does not apply.

Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions. In this case, the fraction does not reduce.

### Example 3

Simplify the expression with exponent

(2x^{-2}y)^2\cdotp (x^2y^{-2})^3

=4x^{-4}y^2\cdotp x^6y^{-6}

=4\dfrac{x^6}{x^4}\cdotp \dfrac{y^2}{y^6}

=\dfrac{4x^2}{y^4}

Step 1: Apply the Zero-Exponent Rule. In this case, there are no zero powers.

Step 2: Apply the Power Rule.

Step 3: Apply the Negative Exponent Rule. Move every negative exponent in the numerator to the denominator and vice versa.

Step 4: Apply the Product Rule. In this case, the product rule does not apply.

Step 5: Apply the Quotient Rule. In this case, the x’s ended up in the numerator and the y’s ended up in the denominator.

Step 6: Raise each coefficient (or number) to the appropriate power and then simplify or reduce any remaining fractions.